Concept explainers
a.
Construct a payoffs table.
a.
Explanation of Solution
On observing the given information, it is clear that the lease about for a complete part set for summer season is $560. The rental amount per day is $5. Profit for each set for the first season is $10.
If the firm leased 41 complete sets, the profit for only 41 sets rented is shown below:
If the firm leased 42 complete sets, the profit for only 41 sets rented is shown below:
In a similar way, the calculations are obtained.
The payoffs are computed as follows:
Act | 41 | 42 | 43 | 44 | 45 | 46 |
41 | $410 | $410 | $410 | $410 | $410 | $410 |
42 | $405 | $420 | $420 | $420 | $420 | $420 |
43 | $400 | $415 | $430 | $430 | $430 | $430 |
44 | $395 | $410 | $425 | $440 | $440 | $440 |
45 | $390 | $405 | $420 | $435 | $450 | $450 |
46 | $385 | $400 | $415 | $430 | $445 | $460 |
b.
Find the expected payoffs for leasing 41, 42, and 44 complete sets from the Boston firms .
Give the recommended decision.
b.
Answer to Problem 17CE
The expected payoffs are given below:
Event | Expected payoffs |
41 | $410 |
42 | $419.1 |
43 | $426.70 |
44 | $432.20 |
45 | $431.70 |
46 | $427.45 |
Explanation of Solution
The expected profit for 41 sets is shown below:
Event | Payoff value | Probability | Expected payoffs |
41 | $410 | ||
42 | $410 | ||
43 | $410 | ||
44 | $410 | ||
45 | $410 | ||
46 | $410 | ||
Expected profit( | $410 |
The expected profit for 42 sets is shown below:
Event | Payoff value | Expected payoffs |
41 | $405 | |
42 | $420 | |
43 | $420 | |
44 | $420 | |
45 | $420 | |
46 | $420 | |
Expected profit( | $419.10 |
The expected profit for 43 sets is shown below:
Event | Payoff value | Expected payoffs |
41 | $400 | |
42 | $415 | |
43 | $430 | |
44 | $430 | |
45 | $430 | |
46 | $430 | |
Expected profit( | $426.70 |
Therefore, the expected profit for all the sets will be obtained in the same way.
The expected profit all the events is shown below:
Event | Expected payoffs |
41 | $410 |
42 | $419.1 |
43 | $426.70 |
44 | $432.20 |
45 | $431.70 |
46 | $427.45 |
c.
Identify the most profitable alternative based on the expected daily profit.
c.
Explanation of Solution
The expected profit for the event
Thus, the event
Hence, order 44 sets because the expected profit of $432.20 is the highest.
d.
Obtain the expected opportunity loss for 41, 42, and 43.
d.
Answer to Problem 17CE
The expected opportunity loss table is obtained as follows:
Act | ||||||
41 | 42 | 43 | 44 | 9 | 45 | |
Expected opp. loss | $28.3 | $19.2 | $11.6 | $6.10 | $6.60 | $10.85 |
Explanation of Solution
Opportunity loss:
From the table in Part (a) if the Supply
Then, the opportunity loss is obtained by taking the difference between $410 and $410 That is,
Thus, the opportunity loss for
In the same way, the remaining values in the table are obtained. If the Supply
Then, the opportunity loss is obtained by taking the difference between $410 and $405, that is,
Thus, the opportunity loss for
The opportunity loss table is obtained as follows:
Supply | Opportunity loss | |||||
41 | 42 | 43 | 44 | 45 | 46 | |
41 | $0 | $10 | $20 | $30 | $40 | $50 |
42 | $5 | $0 | $10 | $20 | $30 | $40 |
43 | $10 | $5 | $0 | $10 | $20 | $30 |
44 | $15 | $10 | $5 | $0 | $10 | $20 |
45 | $20 | $15 | $10 | $5 | $0 | $10 |
46 | $25 | $20 | $15 | $10 | $5 | $0 |
Then, the expected opportunity loss for 41 sets is calculated as follows:
Thus, the expected opportunity loss for 41 sets is $28.3.
The expected opportunity loss for 42 sets is calculated as follows:
Thus, the expected opportunity loss for 42 sets is $19.2.
The expected opportunity loss for 43 sets is calculated as follows:
Thus, the expected opportunity loss for 42 sets is $11.6.
In the same way, the remaining values are obtained. The expected opportunity losses are shown below:
Act | ||||||
41 | 42 | 43 | 44 | 9 | 45 | |
Expected opp. loss | $28.3 | $19.2 | $11.6 | $6.10 | $6.60 | $10.85 |
e.
Obtain the most profitable course of action to take based on the expected opportunity loss table.
State whether the decision made in Part (c) is agreed in this context.
e.
Explanation of Solution
On observing the table in Part (d), the opportunity loss when ordering 44 sets is minimum. Therefore, ordering 44 sets is most profitable.
Both the results stated in Part (c) and Part (e) conclude with the same results. Therefore, the result obtained in Part (c) is agreed with result in this context.
f.
Obtain the expected value of perfect information.
f.
Answer to Problem 17CE
The expected value of perfect information is $6.10.
Explanation of Solution
It is given that
The above calculations show that the EMV for the ordering 44 sets is the maximum at $432.20.
In order to maximize the profit, it is appropriate to order 44.
The expected value of perfect information can be calculated using the following formula:
The expected monetary values calculated above provide the expected values under conditions of uncertainty.
Now, under conditions of certainty, the maximum rent
Using the probabilities, the expected value under conditions of certainty is calculated below:
Hence, the expected value of perfect information is calculated below:
Thus, the expected value of perfect information is $6.10.
On observing the evidence stated in the previous parts, the maximum amount to be paid for the perfect information is $6.10.
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Chapter 20 Solutions
Loose Leaf for Statistical Techniques in Business and Economics
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